Revision date: 03 Sep 2025 · Version: 0.9 · Authors: Mustafa Aksu, ChatGPT (GPT-5 Pro)
Summary. In RTG, local gauge structure emerges only when closed resonance loops are abundant and stable. In the band
\(0.70 \le x \equiv |\Delta\omega|/\Delta\omega^\* < 1.55\) (“4‑D corridors”), the network organizes into elongated,
low‑loop‑density pathways. These support transport and look higher‑dimensional in diffusion tests, but they do not
sustain the holonomy/loop statistics required to promote the exchange term to a local U(1) gauge field. Hence this band is treated
as global (non‑gauged) in the current EFT. See the EFT page:
From RTG to an Effective Field Theory (EFT).
Contents
1 · Window definition and geometric intuition
Let \( x \equiv |\Delta\omega|/\Delta\omega^\* \). Empirically, RTG nodes coupled by resonance bonds form different mesoscale organizations:
(1) U(1) (0 ≤ x < 0.28), (2) soft‑spin SU(2) (0.28 ≤ x ≤ 0.70), (3) the 4‑D corridors
(0.70 ≤ x < 1.55), and (4) a narrow U(1)\(^2\) (Cartan) window (1.55 ≤ x ≤ 1.70).
Intuition for the corridors. Beyond \(x\simeq 0.70\) the resonance Q‑factor drops sharply (“decoherence cliff”).
Closed loops become rare, while long, nearly self‑avoiding paths proliferate. Random‑walk probes see an effective diffusion
dimensionality above 3 (hence “4‑D‑like corridors”), but without the loop density needed for a local gauge structure.
2 · When does gauge emerge?
Our U(1) EFT arises from the gauge‑invariant combination \( \Delta\phi_{ij} – a B_{ij} \) and a small‑angle expansion in which the
quadratic form in \(B\) becomes the Maxwell term (see EFT §3).
For this continuum limit to be self‑consistent on a lattice, three conditions must simultaneously hold:
- Loop abundance. A non‑vanishing density of closed resonance loops across scales \(L\) (so that link
phases define meaningful holonomies on many loops). - Phase regularity. A well‑behaved phase distribution (uniform over \([- \pi,\pi]\) is used) so linear terms
cancel on average and the quadratic form is positive. - Locality. Short‑range link statistics dominate (windowed by \(x\)), so the continuum limit is local
and regulator‑stable.
U(1), SU(2), and U(1)\(^2\) windows satisfy these; the corridors do not (see below).
3 · Why the corridors are not gauged
- Low loop density. The probability \( \rho_{\rm loop}(L) \) to find closed resonance loops of perimeter \(L\)
falls rapidly for \( x \ge 0.70 \). Without loops, a holonomy‑based gauge identification is ill‑posed. - Transport‑dominant geometry. The network is dominated by elongated paths that support conduction/propagation
but offer few contractible cycles. A Wilson‑like loop average shows no stable area law necessary for a Maxwell match. - Phase decorrelation. Near and above the decoherence cliff, phase‑locking weakens; the small‑angle,
phase‑averaged expansion that yields a clean \( \tfrac14 \kappa_B F^2 \) becomes unreliable.
Conclusion: We treat the corridor band as global (non‑gauged) in v1.x of the EFT.
If future diagnostics (below) reverse these signals—i.e., high loop density, stable holonomies, regulator‑robust quadratic matching—
we will promote a gauge description for a sub‑region of this band.
4 · Diagnostics to test the claims
We recommend four quantitative probes. Each can be computed on the same link ensembles you use for \(C_\kappa\).
GVT‑1 · Loop density curve
Estimate the density of closed resonance loops \( \rho_{\rm loop}(L) \) per volume vs loop length \(L\). Acceptance for gauging:
sustained power‑law or slower decay over a decade in \(L\). Corridor expectation: fast decay; orders of magnitude below SU(2)/U(1).
GVT‑2 · Wilson‑like holonomy
Define \( U_{ij} = e^{i(\phi_i-\phi_j)} \) on bonds (omit \(B\) to test the native lattice), then compute
\( W(C)=\big\langle \mathrm{Re}\,\prod_{(ij)\in C} U_{ij} \big\rangle \) on elementary and mesoscopic loops \(C\).
A Maxwell‑capable regime shows a stable, orientation‑averaged area law at small areas. Corridors should fail this stability test.
GVT‑3 · Spectral dimension \( d_s \)
With return probability \(P(t)\) of a simple random walk on the resonance network,
\[
d_s(t) = -2\,\frac{d\ln P(t)}{d\ln t}.
\]
Corridors typically show \( d_s(t) \) above 3 at intermediate \(t\), but this is not sufficient for gauging; combine with GVT‑1/2.
GVT‑4 · Phase‑regularity check
Verify the phase distribution used in the small‑angle expansion is effectively uniform in the corridor band; significant skew
implies the quadratic‑form match to Maxwell is not robust.
Acceptance rule. Consider gauging a sub‑band of the corridors only if:
(i) \( \rho_{\rm loop}(L) \) shows a sustained tail comparable to SU(2),
(ii) a small‑area Wilson‑like average is stable and orientation‑robust,
(iii) the quadratic expansion is regulator‑stable, and
(iv) Ward‑type residuals extrapolate to zero within errors.
5 · Predictions & observables in the corridor band
- Spectral‑dimension uplift: \( d_s(t) \) exceeds 3 over an intermediate diffusion scale.
- Anisotropic transport: higher conduction along elongated paths; fewer transverse loopbacks.
- Suppressed CHSH‑quality correlations: consistent with the decoherence increase near \(x\simeq 0.70\).
- No clean Maxwell matching: attempts to fit \( \tfrac14 \kappa_B F^2 \) are regulator‑sensitive or unstable.
6 · Bridge to the EFT page
The EFT page (link) gauges U(1) (0–0.28), soft‑spin SU(2) (0.28–0.70), and U(1)\(^2\) (1.55–1.70).
It explicitly does not gauge the 0.70–1.55 corridor because GVT‑1..4 fail in our current runs. If future data pass the gates
above for a narrower sub‑window, we will integrate a “Corridor EFT” addendum with its own normalization and Ward tests.
7 · Figures & artifacts
- Corridor x‑histogram: corridor_x_hist.png
- Loop density (GVT‑1): corridor_loop_density.png
- Wilson‑like averages (GVT‑2): corridor_wilson_loops.png
- Spectral dimension \( d_s(t) \) (GVT‑3): corridor_spectral_dimension.png
